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<br> <h1> Link to the source code for this example: </h1>
<a href="https://github.com/libMesh/libmesh/tree/master/examples/adaptivity/adaptivity_ex4" target="_blank">Open adaptivity_ex4 in new tab.</a>
<a name="output"></a> 
<br><br><br> <h1> The console output of the program: </h1> 
<pre>
  CXX      example_dbg-adaptivity_ex4.o
  CXXLD    example-dbg
***************************************************************
* Running Example adaptivity_ex4:
*   example-opt  
***************************************************************
 
 EquationSystems
  n_systems()=1
   System #0, "Biharmonic"
    Type "LinearImplicit"
    Variables="u" 
    Finite Element Types="HERMITE", "JACOBI_20_00" 
    Infinite Element Mapping="CARTESIAN" 
    Approximation Orders="THIRD", "THIRD" 
    n_dofs()=36
    n_local_dofs()=36
    n_constrained_dofs()=0
    n_local_constrained_dofs()=0
    n_vectors()=1
    n_matrices()=1
    DofMap Sparsity
      Average  On-Processor Bandwidth <= 21.7778
      Average Off-Processor Bandwidth <= 0
      Maximum  On-Processor Bandwidth <= 36
      Maximum Off-Processor Bandwidth <= 0
    DofMap Constraints
      Number of DoF Constraints = 0

 Mesh Information:
  elem_dimensions()={2}
  spatial_dimension()=2
  n_nodes()=25
    n_local_nodes()=25
  n_elem()=4
    n_local_elem()=4
    n_active_elem()=4
  n_subdomains()=1
  n_partitions()=1
  n_processors()=1
  n_threads()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System #0, "Biharmonic"
    Type "LinearImplicit"
    Variables="u" 
    Finite Element Types="HERMITE", "JACOBI_20_00" 
    Infinite Element Mapping="CARTESIAN" 
    Approximation Orders="THIRD", "THIRD" 
    n_dofs()=36
    n_local_dofs()=36
    n_constrained_dofs()=0
    n_local_constrained_dofs()=0
    n_vectors()=1
    n_matrices()=1
    DofMap Sparsity
      Average  On-Processor Bandwidth <= 21.7778
      Average Off-Processor Bandwidth <= 0
      Maximum  On-Processor Bandwidth <= 36
      Maximum Off-Processor Bandwidth <= 0
    DofMap Constraints
      Number of DoF Constraints = 0

Beginning Solve 0
Linear solver converged at step: 5, final residual: 2.30574e-13
L2-Norm is: 0.384025
H1-Norm is: 1.98976
H2-Norm is: 14.3417

L2-Error is: 0.0335358
H1-Error is: 0.267039
H2-Error is: 3.51162

  Refining the mesh...
Mean Error: 4.54516e-13
Error Variance: 5.31276e-26
 Mesh Information:
  elem_dimensions()={2}
  spatial_dimension()=2
  n_nodes()=41
    n_local_nodes()=41
  n_elem()=8
    n_local_elem()=8
    n_active_elem()=7
  n_subdomains()=1
  n_partitions()=1
  n_processors()=1
  n_threads()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System #0, "Biharmonic"
    Type "LinearImplicit"
    Variables="u" 
    Finite Element Types="HERMITE", "JACOBI_20_00" 
    Infinite Element Mapping="CARTESIAN" 
    Approximation Orders="THIRD", "THIRD" 
    n_dofs()=56
    n_local_dofs()=56
    n_constrained_dofs()=8
    n_local_constrained_dofs()=8
    n_vectors()=1
    n_matrices()=1
    DofMap Sparsity
      Average  On-Processor Bandwidth <= 27.4286
      Average Off-Processor Bandwidth <= 0
      Maximum  On-Processor Bandwidth <= 52
      Maximum Off-Processor Bandwidth <= 0
    DofMap Constraints
      Number of DoF Constraints = 8
      Average DoF Constraint Length= 4

Beginning Solve 1
Linear solver converged at step: 5, final residual: 4.32422e-12
L2-Norm is: 0.385688
H1-Norm is: 1.99188
H2-Norm is: 14.3773

L2-Error is: 0.0312488
H1-Error is: 0.251755
H2-Error is: 3.36347

  Refining the mesh...
Mean Error: 0.904773
Error Variance: 0.231378
 Mesh Information:
  elem_dimensions()={2}
  spatial_dimension()=2
  n_nodes()=69
    n_local_nodes()=69
  n_elem()=16
    n_local_elem()=16
    n_active_elem()=13
  n_subdomains()=1
  n_partitions()=1
  n_processors()=1
  n_threads()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System #0, "Biharmonic"
    Type "LinearImplicit"
    Variables="u" 
    Finite Element Types="HERMITE", "JACOBI_20_00" 
    Infinite Element Mapping="CARTESIAN" 
    Approximation Orders="THIRD", "THIRD" 
    n_dofs()=88
    n_local_dofs()=88
    n_constrained_dofs()=8
    n_local_constrained_dofs()=8
    n_vectors()=1
    n_matrices()=1
    DofMap Sparsity
      Average  On-Processor Bandwidth <= 28
      Average Off-Processor Bandwidth <= 0
      Maximum  On-Processor Bandwidth <= 52
      Maximum Off-Processor Bandwidth <= 0
    DofMap Constraints
      Number of DoF Constraints = 8
      Average DoF Constraint Length= 4

Beginning Solve 2
Linear solver converged at step: 8, final residual: 4.96983e-12
L2-Norm is: 0.39603
H1-Norm is: 2.01114
H2-Norm is: 14.5761

L2-Error is: 0.020273
H1-Error is: 0.166208
H2-Error is: 2.36714

  Refining the mesh...
Mean Error: 1.07284
Error Variance: 1.26032
 Mesh Information:
  elem_dimensions()={2}
  spatial_dimension()=2
  n_nodes()=125
    n_local_nodes()=125
  n_elem()=32
    n_local_elem()=32
    n_active_elem()=25
  n_subdomains()=1
  n_partitions()=1
  n_processors()=1
  n_threads()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System #0, "Biharmonic"
    Type "LinearImplicit"
    Variables="u" 
    Finite Element Types="HERMITE", "JACOBI_20_00" 
    Infinite Element Mapping="CARTESIAN" 
    Approximation Orders="THIRD", "THIRD" 
    n_dofs()=152
    n_local_dofs()=152
    n_constrained_dofs()=32
    n_local_constrained_dofs()=32
    n_vectors()=1
    n_matrices()=1
    DofMap Sparsity
      Average  On-Processor Bandwidth <= 33.6842
      Average Off-Processor Bandwidth <= 0
      Maximum  On-Processor Bandwidth <= 60
      Maximum Off-Processor Bandwidth <= 0
    DofMap Constraints
      Number of DoF Constraints = 32
      Average DoF Constraint Length= 4

Beginning Solve 3
Linear solver converged at step: 12, final residual: 1.95735e-12
L2-Norm is: 0.405284
H1-Norm is: 2.03173
H2-Norm is: 14.7515

L2-Error is: 0.00185729
H1-Error is: 0.0267654
H2-Error is: 0.71977

  Refining the mesh...
Mean Error: 0.210827
Error Variance: 0.0183284
 Mesh Information:
  elem_dimensions()={2}
  spatial_dimension()=2
  n_nodes()=249
    n_local_nodes()=249
  n_elem()=68
    n_local_elem()=68
    n_active_elem()=52
  n_subdomains()=1
  n_partitions()=1
  n_processors()=1
  n_threads()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System #0, "Biharmonic"
    Type "LinearImplicit"
    Variables="u" 
    Finite Element Types="HERMITE", "JACOBI_20_00" 
    Infinite Element Mapping="CARTESIAN" 
    Approximation Orders="THIRD", "THIRD" 
    n_dofs()=292
    n_local_dofs()=292
    n_constrained_dofs()=72
    n_local_constrained_dofs()=72
    n_vectors()=1
    n_matrices()=1
    DofMap Sparsity
      Average  On-Processor Bandwidth <= 35.4521
      Average Off-Processor Bandwidth <= 0
      Maximum  On-Processor Bandwidth <= 76
      Maximum Off-Processor Bandwidth <= 0
    DofMap Constraints
      Number of DoF Constraints = 72
      Average DoF Constraint Length= 4

Beginning Solve 4
Linear solver converged at step: 23, final residual: 2.1448e-12
L2-Norm is: 0.405907
H1-Norm is: 2.03286
H2-Norm is: 14.7613

L2-Error is: 0.00113017
H1-Error is: 0.016604
H2-Error is: 0.482316

  Refining the mesh...
Mean Error: 0.108787
Error Variance: 0.00898017
 Mesh Information:
  elem_dimensions()={2}
  spatial_dimension()=2
  n_nodes()=485
    n_local_nodes()=485
  n_elem()=140
    n_local_elem()=140
    n_active_elem()=106
  n_subdomains()=1
  n_partitions()=1
  n_processors()=1
  n_threads()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System #0, "Biharmonic"
    Type "LinearImplicit"
    Variables="u" 
    Finite Element Types="HERMITE", "JACOBI_20_00" 
    Infinite Element Mapping="CARTESIAN" 
    Approximation Orders="THIRD", "THIRD" 
    n_dofs()=548
    n_local_dofs()=548
    n_constrained_dofs()=120
    n_local_constrained_dofs()=120
    n_vectors()=1
    n_matrices()=1
    DofMap Sparsity
      Average  On-Processor Bandwidth <= 36.7007
      Average Off-Processor Bandwidth <= 0
      Maximum  On-Processor Bandwidth <= 68
      Maximum Off-Processor Bandwidth <= 0
    DofMap Constraints
      Number of DoF Constraints = 120
      Average DoF Constraint Length= 4

Beginning Solve 5
Linear solver converged at step: 37, final residual: 2.04494e-11
L2-Norm is: 0.406259
H1-Norm is: 2.03204
H2-Norm is: 14.7673

L2-Error is: 0.000356962
H1-Error is: 0.00613803
H2-Error is: 0.227619

  Refining the mesh...
Mean Error: 0.0264705
Error Variance: 0.00117171
 Mesh Information:
  elem_dimensions()={2}
  spatial_dimension()=2
  n_nodes()=957
    n_local_nodes()=957
  n_elem()=280
    n_local_elem()=280
    n_active_elem()=211
  n_subdomains()=1
  n_partitions()=1
  n_processors()=1
  n_threads()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System #0, "Biharmonic"
    Type "LinearImplicit"
    Variables="u" 
    Finite Element Types="HERMITE", "JACOBI_20_00" 
    Infinite Element Mapping="CARTESIAN" 
    Approximation Orders="THIRD", "THIRD" 
    n_dofs()=1072
    n_local_dofs()=1072
    n_constrained_dofs()=304
    n_local_constrained_dofs()=304
    n_vectors()=1
    n_matrices()=1
    DofMap Sparsity
      Average  On-Processor Bandwidth <= 38.4776
      Average Off-Processor Bandwidth <= 0
      Maximum  On-Processor Bandwidth <= 76
      Maximum Off-Processor Bandwidth <= 0
    DofMap Constraints
      Number of DoF Constraints = 304
      Average DoF Constraint Length= 4

Beginning Solve 6
Linear solver converged at step: 294, final residual: 8.22354e-11
L2-Norm is: 0.406309
H1-Norm is: 2.03185
H2-Norm is: 14.7684

L2-Error is: 8.72883e-05
H1-Error is: 0.00241104
H2-Error is: 0.13224

  Refining the mesh...
Mean Error: 0.0109873
Error Variance: 0.000156803
 Mesh Information:
  elem_dimensions()={2}
  spatial_dimension()=2
  n_nodes()=1885
    n_local_nodes()=1885
  n_elem()=560
    n_local_elem()=560
    n_active_elem()=421
  n_subdomains()=1
  n_partitions()=1
  n_processors()=1
  n_threads()=1
  processor_id()=0

 EquationSystems
  n_systems()=1
   System #0, "Biharmonic"
    Type "LinearImplicit"
    Variables="u" 
    Finite Element Types="HERMITE", "JACOBI_20_00" 
    Infinite Element Mapping="CARTESIAN" 
    Approximation Orders="THIRD", "THIRD" 
    n_dofs()=2088
    n_local_dofs()=2088
    n_constrained_dofs()=616
    n_local_constrained_dofs()=616
    n_vectors()=1
    n_matrices()=1
    DofMap Sparsity
      Average  On-Processor Bandwidth <= 38.9885
      Average Off-Processor Bandwidth <= 0
      Maximum  On-Processor Bandwidth <= 72
      Maximum Off-Processor Bandwidth <= 0
    DofMap Constraints
      Number of DoF Constraints = 616
      Average DoF Constraint Length= 4

Beginning Solve 7
Linear solver converged at step: 1000, final residual: 2.62662e-07
L2-Norm is: 0.406333
H1-Norm is: 2.03181
H2-Norm is: 14.7688

L2-Error is: 5.02524e-05
H1-Error is: 0.00140302
H2-Error is: 0.082078


 -------------------------------------------------------------------------------------------------------------------
| Time:           Wed Apr 13 09:22:42 2016                                                                          |
| OS:             Darwin                                                                                            |
| HostName:       peterson-laptop.local                                                                             |
| OS Release:     15.0.0                                                                                            |
| OS Version:     Darwin Kernel Version 15.0.0: Sat Sep 19 15:53:46 PDT 2015; root:xnu-3247.10.11~1/RELEASE_X86_64  |
| Machine:        x86_64                                                                                            |
| Username:       petejw                                                                                            |
| Configuration:  ../configure  'INSTALL=/usr/bin/install -C'                                                       |
|  '--with-methods=opt dbg'                                                                                         |
|  '--prefix=/Users/petejw/projects/libmesh_git/installed'                                                          |
|  '--enable-perflog'                                                                                               |
|  '--enable-tracefiles'                                                                                            |
|  '--disable-blocked-storage'                                                                                      |
|  '--enable-default-comm-world'                                                                                    |
|  '--enable-tecio'                                                                                                 |
|  '--enable-unique-ptr'                                                                                            |
|  '--enable-ifem'                                                                                                  |
|  'PETSC_DIR=/opt/moose/petsc/mpich_petsc-3.6.1/clang-opt-superlu'                                                 |
|  'SLEPC_DIR=/opt/petejw/slepc/3.6.1'                                                                              |
|  'CPPFLAGS= '                                                                                                     |
|  'LIBS= '                                                                                                         |
|  'CPPFLAGS=  '                                                                                                    |
|  'LIBS=  '                                                                                                        |
|  '--disable-netcdf-4'                                                                                             |
|  '--disable-testsets'                                                                                             |
|  'CXX=mpicxx'                                                                                                     |
|  'CC=mpicc'                                                                                                       |
|  'F77=mpif77'                                                                                                     |
|  'FC=mpif90'                                                                                                      |
|  'CPPFLAGS=   '                                                                                                   |
|  'LIBS=   '                                                                                                       |
 -------------------------------------------------------------------------------------------------------------------
 -----------------------------------------------------------------------------------------------------------------
| libMesh Performance: Alive time=2.98852, Active time=2.94658                                                    |
 -----------------------------------------------------------------------------------------------------------------
| Event                              nCalls     Total Time  Avg Time    Total Time  Avg Time    % of Active Time  |
|                                               w/o Sub     w/o Sub     With Sub    With Sub    w/o S    With S   |
|-----------------------------------------------------------------------------------------------------------------|
|                                                                                                                 |
|                                                                                                                 |
| DofMap                                                                                                          |
|   add_neighbors_to_send_list()     8          0.0004      0.000049    0.0004      0.000049    0.01     0.01     |
|   build_constraint_matrix()        835        0.0027      0.000003    0.0027      0.000003    0.09     0.09     |
|   build_sparsity()                 8          0.0089      0.001106    0.0093      0.001157    0.30     0.31     |
|   cnstrn_elem_mat_vec()            835        0.0046      0.000005    0.0046      0.000005    0.16     0.16     |
|   create_dof_constraints()         8          0.0203      0.002532    0.2634      0.032923    0.69     8.94     |
|   distribute_dofs()                8          0.0005      0.000061    0.0020      0.000256    0.02     0.07     |
|   dof_indices()                    9115       0.0034      0.000000    0.0034      0.000000    0.12     0.12     |
|   enforce_constraints_exactly()    7          0.0005      0.000065    0.0005      0.000065    0.02     0.02     |
|   old_dof_indices()                3617       0.0015      0.000000    0.0015      0.000000    0.05     0.05     |
|   prepare_send_list()              9          0.0000      0.000000    0.0000      0.000000    0.00     0.00     |
|   reinit()                         8          0.0016      0.000195    0.0016      0.000195    0.05     0.05     |
|                                                                                                                 |
| EquationSystems                                                                                                 |
|   build_solution_vector()          1          0.0584      0.058436    0.0586      0.058589    1.98     1.99     |
|                                                                                                                 |
| ErrorVector                                                                                                     |
|   mean()                           14         0.0000      0.000001    0.0000      0.000001    0.00     0.00     |
|   variance()                       7          0.0000      0.000001    0.0000      0.000001    0.00     0.00     |
|                                                                                                                 |
| ExodusII_IO                                                                                                     |
|   write_nodal_data()               1          0.0017      0.001678    0.0017      0.001678    0.06     0.06     |
|                                                                                                                 |
| FE                                                                                                              |
|   compute_shape_functions()        12104      0.3042      0.000025    0.3042      0.000025    10.32    10.32    |
|   init_shape_functions()           12104      2.0802      0.000172    2.0802      0.000172    70.60    70.60    |
|   inverse_map()                    31968      0.0210      0.000001    0.0210      0.000001    0.71     0.71     |
|                                                                                                                 |
| FEMap                                                                                                           |
|   compute_affine_map()             14507      0.0156      0.000001    0.0156      0.000001    0.53     0.53     |
|   compute_face_map()               3134       0.0062      0.000002    0.0158      0.000005    0.21     0.54     |
|   init_face_shape_functions()      3134       0.0015      0.000000    0.0015      0.000000    0.05     0.05     |
|   init_reference_to_physical_map() 12104      0.0429      0.000004    0.0429      0.000004    1.46     1.46     |
|                                                                                                                 |
| GenericProjector                                                                                                |
|   copy_dofs                        279        0.0002      0.000001    0.0008      0.000003    0.01     0.03     |
|   operator()                       7          0.0018      0.000251    1.1120      0.158863    0.06     37.74    |
|   project_edges                    556        0.0000      0.000000    0.0000      0.000000    0.00     0.00     |
|   project_interior                 556        0.0001      0.000000    0.0001      0.000000    0.00     0.00     |
|   project_nodes                    556        0.0041      0.000007    1.1079      0.001993    0.14     37.60    |
|   project_sides                    556        0.0006      0.000001    0.0006      0.000001    0.02     0.02     |
|                                                                                                                 |
| JumpErrorEstimator                                                                                              |
|   estimate_error()                 7          0.0135      0.001923    1.1196      0.159941    0.46     38.00    |
|                                                                                                                 |
| Mesh                                                                                                            |
|   all_second_order()               1          0.0000      0.000036    0.0000      0.000036    0.00     0.00     |
|   contract()                       7          0.0001      0.000007    0.0001      0.000014    0.00     0.00     |
|   find_neighbors()                 9          0.0020      0.000219    0.0020      0.000219    0.07     0.07     |
|   renumber_nodes_and_elem()        25         0.0001      0.000005    0.0001      0.000005    0.00     0.00     |
|                                                                                                                 |
| MeshOutput                                                                                                      |
|   write_equation_systems()         1          0.0000      0.000013    0.0603      0.060281    0.00     2.05     |
|                                                                                                                 |
| MeshRefinement                                                                                                  |
|   _coarsen_elements()              14         0.0000      0.000002    0.0000      0.000002    0.00     0.00     |
|   _refine_elements()               14         0.0022      0.000157    0.0039      0.000275    0.07     0.13     |
|   add_node()                       5004       0.0012      0.000000    0.0012      0.000000    0.04     0.04     |
|   make_coarsening_compatible()     15         0.0006      0.000039    0.0006      0.000039    0.02     0.02     |
|   make_flags_parallel_consistent() 21         0.0003      0.000013    0.0003      0.000013    0.01     0.01     |
|   make_refinement_compatible()     15         0.0001      0.000005    0.0001      0.000005    0.00     0.00     |
|                                                                                                                 |
| MeshTools::Generation                                                                                           |
|   build_cube()                     1          0.0000      0.000037    0.0000      0.000037    0.00     0.00     |
|                                                                                                                 |
| OldSolutionValue                                                                                                |
|   Gradient eval_at_node()          2224       0.0005      0.000000    0.0005      0.000000    0.02     0.02     |
|   Number eval_at_node()            2224       0.0002      0.000000    0.0002      0.000000    0.01     0.01     |
|   check_old_context(c)             279        0.0002      0.000001    0.0005      0.000002    0.01     0.02     |
|   check_old_context(c,p)           7440       0.0045      0.000001    0.0102      0.000001    0.15     0.35     |
|   eval_at_point()                  7440       0.1817      0.000024    1.1022      0.000148    6.17     37.40    |
|   eval_old_dofs()                  279        0.0001      0.000000    0.0006      0.000002    0.00     0.02     |
|                                                                                                                 |
| Parallel                                                                                                        |
|   allgather()                      8          0.0000      0.000000    0.0000      0.000000    0.00     0.00     |
|                                                                                                                 |
| Partitioner                                                                                                     |
|   single_partition()               9          0.0001      0.000009    0.0001      0.000009    0.00     0.00     |
|                                                                                                                 |
| PetscLinearSolver                                                                                               |
|   solve()                          8          0.1392      0.017394    0.1392      0.017394    4.72     4.72     |
|                                                                                                                 |
| System                                                                                                          |
|   assemble()                       8          0.0124      0.001545    0.1201      0.015007    0.42     4.07     |
|   project_vector(old,new)          7          0.0013      0.000192    1.1143      0.159188    0.05     37.82    |
|                                                                                                                 |
| TopologyMap                                                                                                     |
|   init()                           14         0.0037      0.000262    0.0037      0.000262    0.12     0.12     |
 -----------------------------------------------------------------------------------------------------------------
| Totals:                            131120     2.9466                                          100.00            |
 -----------------------------------------------------------------------------------------------------------------

 
***************************************************************
* Done Running Example adaptivity_ex4:
*   example-opt  
***************************************************************
</pre>
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